Missing variable problems

Transcription

1 Missing variable problems In many vision problems, if some variables were known the maximum likelihood inference problem would be easy fitting; if we knew which line each token came from, it would be easy to determine line parameters segmentation; if we knew the segment each pixel came from, it would be easy to determine the segment parameters fundamental matrix estimation; if we knew which feature corresponded to which, it would be easy to determine the fundamental matrix etc. This sort of thing happens in statistics, too

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3 For Independent data samples

4 Missing variables - strategy We have a problem with parameters, missing variables This suggests: Iterate until convergence replace missing variable with expected values, given fixed values of parameters fix missing variables, choose parameters to maximise likelihood given fixed values of missing variables e.g., iterate till convergence allocate each point to a line with a weight, which is the probability of the point given the line refit lines to the weighted set of points Converges to local extremum

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6 Finite Mixtures P(x) = S i=1:3 a(i) g i (x; q)

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11 Expection on Indicator variables

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14 Segmentation with EM Figure from Color and Texture Based Image Segmentation Using EM and Its Application to Content Based Image Retrieval,S.J. Belongie et al., Proc. Int. Conf. Computer Vision, 1998, c1998, IEEE

15 We have one line, and n points Some come from the line, some from noise This is a mixture model: P point line and noise params Lines and robustness ( ) = P point line We wish to determine line parameters p(comes from line) ( )P( comes from line)+ P( point noise)p( comes from noise) = P( point line)l + P( point noise)(1- l)

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17 Estimating the mixture model Introduce a set of hidden variables, d, one for each point. They are one when the point is on the line, and zero when off. If these are known, the negative log-likelihood becomes (the line s parameters are f, c): Here K is a normalising constant, kn is the noise intensity (we ll choose this later). Q c ( x;q ) = Â i Ê Á Á Ë d i Ê Á Ë ( x i cosf + y i sinf + c) 2 2s 2 ( 1- d i )k n ˆ + ˆ + K

18 Substituting for delta We shall substitute the expected value of d, for a given q recall q=(f, c, l) E(d_i)=1. P(d_i=1 q)+0... Notice that if kn is small and positive, then if distance is small, this value is close to 1 and if it is large, close to zero P( d i = 1 q, x i ) = = P( x i d i = 1,q)P( d i = 1) P( x i d i = 1,q)P( d i = 1)+ P( x i d i = 0,q)P d i = 0 ( exp -1 2s 2 [ x i cosf + y i sinj + c] 2 )l ( exp -1 2s 2 [ x i cosf + y i sinj + c] 2 )l + exp -k n ( ) ( ) 1 - l ( )

19 Algorithm for line fitting Obtain some start point q 0 ( ( ) ) ( ) = f ( 0 ),c ( 0),l 0 Now compute d s using formula above Iterate to convergence Now compute maximum likelihood estimate of q ( 1) f, c come from fitting to weighted points l comes by counting

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21 The expected values of the deltas at the maximum (notice the one value close to zero).

22 Closeup of the fit

23 Choosing parameters What about the noise parameter, and the sigma for the line? several methods from first principles knowledge of the problem (seldom really possible) play around with a few examples and choose (usually quite effective, as precise choice doesn t matter much) notice that if kn is large, this says that points very seldom come from noise, however far from the line they lie usually biases the fit, by pushing outliers into the line rule of thumb; its better to fit to the better fitting points, within reason; if this is hard to do, then the model could be a problem

24 Other examples Segmentation a segment is a gaussian that emits feature vectors (which could contain colour; or colour and position; or colour, texture and position). segment parameters are mean and (perhaps) covariance if we knew which segment each point belonged to, estimating these parameters would be easy rest is on same lines as fitting line Fitting multiple lines rather like fitting one line, except there are more hidden variables easiest is to encode as an array of hidden variables, which represent a table with a one where the i th point comes from the j th line, zeros otherwise rest is on same lines as above

25 Issues with EM Local maxima can be a serious nuisance in some problems no guarantee that we have reached the right maximum Starting k means to cluster the points is often a good idea

26 Local maximum

27 which is an excellent fit to some points

28 and the deltas for this maximum

29 A dataset that is well fitted by four lines

30 Result of EM fitting, with one line (or at least, one available local maximum).

31 Result of EM fitting, with two lines (or at least, one available local maximum).

32 Seven lines can produce a rather logical answer

33 Motion segmentation with EM Model image pair (or video sequence) as consisting of regions of parametric motion affine motion is popular Ê Á Ë v x v y ˆ = Ê Á a Ë c Now we need to determine which pixels belong to which region estimate parameters b ˆ d Ê Á xˆ Ê Ë y + t xˆ Á Ë t y Likelihood assume I( x, y,t) = I( x + v x, y + v y,t +1) +noise Straightforward missing variable problem, rest is calculation

34 Three frames from the MPEG flower garden sequence Figure from Representing Images with layers,, by J. Wang and E.H. Adelson, IEEE Transactions on Image Processing, 1994, c 1994, IEEE

35 Grey level shows region no. with highest probability Segments and motion fields associated with them Figure from Representing Images with layers,, by J. Wang and E.H. Adelson, IEEE Transactions on Image Processing, 1994, c 1994, IEEE

36 If we use multiple frames to estimate the appearance of a segment, we can fill in occlusions; so we can re-render the sequence with some segments removed. Figure from Representing Images with layers,, by J. Wang and E.H. Adelson, IEEE Transactions on Image Processing, 1994, c 1994, IEEE

37 Some generalities Many, but not all problems that can be attacked with EM can also be attacked with RANSAC need to be able to get a parameter estimate with a manageably small number of random choices. RANSAC is usually better Didn t present in the most general form in the general form, the likelihood may not be a linear function of the missing variables in this case, one takes an expectation of the likelihood, rather than substituting expected values of missing variables

38 Model Selection We wish to choose a model to fit to data e.g. is it a line or a circle? e.g is this a perspective or orthographic camera? e.g. is there an aeroplane there or is it noise? Issue In general, models with more parameters will fit a dataset better, but are poorer at prediction This means we can t simply look at the negative loglikelihood (or fitting error)

39 Top is not necessarily a better fit than bottom (actually, almost always worse)

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41 We can discount the fitting error with some term in the number of parameters in the model.

42 Discounts AIC (an information criterion) choose model with smallest value of -2L( D;q * )+ 2 p p is the number of parameters BIC (Bayes information criterion) choose model with smallest value of -2L( D;q * )+ p log N N is the number of data points Minimum description length same criterion as BIC, but derived in a completely different way

43 Cross-validation Split data set into two pieces, fit to one, and compute negative log-likelihood on the other Average over multiple different splits Choose the model with the smallest value of this average The difference in averages for two different models is an estimate of the difference in KL divergence of the models from the source of the data

44 Model averaging Very often, it is smarter to use multiple models for prediction than just one e.g. motion capture data there are a small number of schemes that are used to put markers on the body given we know the scheme S and the measurements D, we can estimate the configuration of the body X We want ( ) = P X S 1, D P X D ( )P( S 1 D)+ P( X S 2, D)P( S 2 D) + P( X S 3, D)P( S 3 D) If it is obvious what the scheme is from the data, then averaging makes little difference If it isn t, then not averaging underestimates the variance of X --- we think we have a more precise estimate than we do.

45 Learning as Model fitting

46 Model Fitting, Learning Model: Mapping between observations and world properties. P(o,s) Parametric: P(o,s;ø) where ø are parameters that need to be fit (learned) from observations (data) Density Estimation: Learn P(o,s) from examples. E.g. Histogram, KDE Mixture of parametric densities.

47 What s the problem? Two sets of hidden parameters Goal, find argmax(s) P(s o,ø) But P(s o;ø) depends on ø, thus different ø, different inference of s. Optimize ø? Yes, but optimal ø only good for one s. Need to optimize both at the same time. One Solution: EM algorithm

48 Learning templates for categories Assume a set of labelled examples for stimuli from categories C = 1, 2, etc Stimuli are fixed images in gaussian noise s = I1 + N Then labelled examples can be used to learn a parametric model P(s C) = N(m,K). Let K = s 2 I. How do we estimate m across trials?

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51 EM algorithm Normal maximum likelihood Y unknown? What to do with Iterate, 1) set value for average over Y 2) maximize, use as next value 1) 2)

52 Density Est: Mixture of Densities e.g. q = { m, s }

53 Mixture Model Applications

54 Motion Estimation

55 Motion Estimation

56 Motion Estimation